Mock AIME II 2012 Problems/Problem 11
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Problem
There exist real values of and
such that
,
, and
for some value of
. Let
be the sum of all possible values of
. Find
.
Solution
First, if , then
. We now assume that
.
Now, note that
.
Also, we have
.
Next,
. But we know
, so
.
Since the only possible values of are
and
, our final answer is
.
(It is easy to check that there exists satisfying the equations.)